A convolution back projection algorithm for local · PDF fileA CONVOLUTION BACK PROJECTION ALGORITHM FOR ... Serial-fee code 1446-8735/05 341. ... A convolution back projection algorithm

ANZIAM J.46(2005), 341–360

A CONVOLUTION BACK PROJECTION ALGORITHM FORLOCAL TOMOGRAPHY

CHALLA S. SASTRY1 and P. C. DAS2

(Received 28 July, 2003; revised 15 February, 2004)

Abstract

The present work deals with the problem of recovering a local image from localisedprojections using the concept of approximation identity. It is based on the observation thatthe Hilbert transform of an approximation identity taken from a certain class of compactlysupported functions with sufficiently many zero moments has no significant spread ofsupport. The associated algorithm uses data pertaining to the local region along with asmall amount of data from its vicinity. The main features of the algorithm are simplicityand similarity with standard filtered back projection (FBP) along with the economic use ofdata.

1. Introduction

The problem of recovering a local image from localised projections in X-ray tomog-raphy has acquired importance so that the radiation dosage as well as the time ofexposure can be reduced. The problems relating to uniqueness, well-posednessetc.associated with local inversion of the Radon transform are well known [14]. Recently,wavelet-based methods have proved quite promising for the purpose of Region-Of-Interest (ROI) tomography [7, 15]. One method based on angular harmonics [18]uses projections, exponentially sampled in radial directions, whereby finer samplingis achieved in the ROI and coarser sampling outside it. Nonetheless, this is not a localreconstruction in the true sense, as full length projections are required along certaindirections and the computations involve interpolations. Several other wavelet-basedmethods are also available [5, 7, 15], which use a reasonably small amount of neigh-bourhood data in addition to ROI data. The amount of additional data depends on the

1Artificial Intelligence Lab, Department of Computer and Information Sciences, University ofHyderabad, Hyderabad 500046, India; e-mail:[email protected] of Mathematics, Indian Institute of Technology, Kanpur 208016, India; e-mail:[email protected]© Australian Mathematical Society 2005, Serial-fee code 1446-8735/05

341

http://www.austms.org.au/Publ/ANZIAM/V46P3/2143.html

mailto:[email protected]

mailto:[email protected]

342 Challa S. Sastry and P. C. Das [2]

choice of wavelets used.The aim of the present work is to propose a filtered back projection (FBP)-type

formula for ROI reconstruction using the concept of an approximation identity. Thework of Riederet al. [16, 17] is conceptually related to the present work although thepurpose and details are different. The proposed algorithm, using an approximationidentity taken from a class of compactly supported functions, has the following mainadvantages: (i) it does not use exponential radial sampling and is therefore simpleto implement; (ii) it allows uniform exposure at all angles as in [15]; and (iii) itis analogous to the standard FBP algorithm and hence can use several readymadewell-developed procedures for implementations [2].

The present work is organised into the following six sections. In Section2, wepresent some standard notation, the basic reconstruction technique and make someobservations about the nonlocality of the Hilbert transform. In Section3, we considerthe local reconstruction technique using the concept of an approximation identity.Since the idea is simple, we have tried to derive the representation formula directlywithout referring to any other technical notation. Section4 deals with the errorestimate depending upon the parameter of the approximation identity and the zeromoments satisfied by the corresponding function. In Section5, we construct someexamples of the approximation identity. In Section6, we study the error arising fromthe use of localised data from interior regions. Section7 deals with a comparison ofthe results of this paper with those of Faridaniet al. [8, 9, 10], Riederet al. [16, 17],A. K. Louis [11] and other wavelet-based methods [5, 7, 15]. In the last section, weconsider some simulation results indicating how to use the right parameters.

2. Preliminaries and the reconstruction techniques

Before developing the reconstruction algorithm, we state some basic definitionsand provide information regarding the Hilbert transform, to be used subsequently.

2.1. Basic reconstruction formula Let L2.R2/ and HÞ.R2/ denote respectivelythe space of all square integrable functions and the Sobolev space of orderÞ onR2.Then the norm of the functionf ∈ H Þ.R2/ [1] is given by

‖ f ‖H Þ.R2/ =(∫

R2

| f .!/|2.1 + ‖!‖2/Þ d!

)1=2

< ∞: (2.1)

The Fourier transform off is defined by

f .!/ =∫Rn

f .x/e−2³ i 〈x;!〉 dx: (2.2)

[3] A convolution back projection algorithm for local tomography 343

Here〈x; !〉 stands for the inner product ofx and! inR2. The Radon transform.R� f /of a sufficiently regular functionf onR2 is given for� ∈ [0;2³/ andt ∈ .−∞;∞/

by the formula

.R� f /.t/ =∫R

f .tu� + sv�/ds; (2.3)

whereu� = .cos�; sin�/ andv� = u�+³=2.The basic objective in tomography is to reconstruct from the projection dataR� f

the density functionf : R2 → R having support in a disc of radiusR centred atthe origin inR2. The basic reconstruction technique is based on the following resultcalled the slice theorem [14]:

\.R� f /.!/ = f .!u� /; (2.4)

expressing the Fourier transform off in terms of the Fourier transforms of.R� f / for06 � < 2³ .

Using the Fourier inversion formula in polar form and the slice theorem, thefunction f is reconstructed from the Radon projection data.R� f / by

f .x/ =∫ ³

0

∫R

\.R� f /.!/ei 2³!〈x;u� 〉|!| d!d�: (2.5)

Equation (2.5) is called the back projection formula.

2.2. Effect of nonlocality In (2.5) the Fourier transform ofR� f is multiplied by|!|.In the spatial domain this product can be written in terms of the Hilbert transform asfollows:

IFT(| · |\.R� f /

).t/ = [email protected]� f /.t/ := .3� f /.t/: (2.6)

In fact, for any sufficiently regular univariable functiong we have

H@g.�/ = .3g/.�/: (2.7)

It can be seen that

[.Hg/.!/ = sgn.!/

ig.!/:

Hence the Fourier transform of a function with nonzero average value has discontinuityat the origin. The presence of discontinuity at the origin in the frequency domain hasthe effect of spreading the support of the Hilbert transform of a compactly supportedfunction [14]. However, the essential support of the filtered function does not spreadif the function has a sufficient number of zero moments [15, 14]. Next we come tothe main reconstruction formula of the present paper.


3. CBP based on an approximation identity

We start with the following simple lemma on an approximation identity (see forexample [3, Theorem 2.8]).

LEMMA 3.1. Let� be a compactly supported continuous function such that∫R

�.t/dt = 1

and letg ∈ L1.R/, then at any pointx of continuity ofg

limj →∞

.g ? � j /.x/ = g.x/; where � j = 2 j�.2j /:

Now we recollect that3� f stands [email protected]� f /. In view of the above lemma, atthe points of continuity ‘t ’ of .3� f /

limj →∞

[.3� f / ? � j ].t/ = .3� f /.t/:

Using the definitions given in (2.6) and (2.7), we obtain the following importantidentity, which is basic to our later considerations.

[.3� f / ? � j ] = IFT[\.R� f /| · |� j

]= .R� f / ? IFT.| · |� j / = .R� f / ? .3� j /: (3.1)

In the above, IFT stands for the inverse Fourier transform operation. From the standardback projection formula and (3.1), we have the following approximate reconstructionformula for sufficiently largeJ:

f .x/ =∫ ³

0

.3� f /.〈x;u� 〉/d� ≈∫ ³

0

[.3� f / ? �J ].〈x;u�〉/d�

=∫ ³

0

[.R� f / ? .3�J /].〈x;u�〉/d�

= 2J

∫ ³

0

[.R� f / ? .3�/J ].〈x;u�〉/d� = f�;J.x/ (say): (3.2)

Here we have used the easily verifiable relations.3�J/.s/ = 22J.3�/.2J s/ andgJ.s/ = 2Jg.2Js/. We note thatf�; j .x/ stands for

f�; j .x/ = 2j

∫ ³

0

[.R� f / ? .3�/ j ].〈x;u�〉/d�: (3.3)


REMARK 1. Since the function� is a mollifier, its scale parameterJ acts as aregularisation parameter and the method in (3.2) is a regularisation method. Whenthe function� has no significant spread of support after ramp filtering, the aboveformula gives a local reconstruction procedure. Although taking larger values forJreduces the excess (outside the ROI) data commensurately, it also results in spreadingthe support of the ramp function and the consequent magnification of noise due tothe ramp function. On the other hand taking smaller values forJ leads to higherapproximation errors in (3.2) in the act of approximating.3� f / by .3� f / ? �J . Inview of this, a balanced value ofJ has to be used. We discuss this point whileanalysing simulation results.

REMARK 2. Since �.0/ = 1, the action of the approximation identity onf mayresult in the low-bandpass-filtered version off . Now, suppose that is a compactlysupported, smooth function such that .0/ = 0. Then the function� = � + ½ stillacts as an approximation identity and hence� in (3.2) can be replaced by�. Theparameter½ may be used as a control parameter for the enhancement of the image.Inserting� in place of� in (3.2) and using the linearity of the convolution operationwe get

f�;J.x/ = 2J

∫ ³

0

[.R� f / ? .3�/J ].〈x;u� 〉/d�

+ ½2J

∫ ³

0

[.R� f / ? .3 /J ].〈x;u�〉/d�

= f�;J.x/ + ½ f ;J.x/: (3.4)

The second part in (3.4) may be regarded as a high-bandpass-filtered version off .So far the choices of and� have been independent. In fact, it is possible to chooseseveral ’s like 1; 2; : : : ; p and get a more generalised expression for (3.4).

When the function considered is oscillatory and orthogonal to�, that is,〈�; 〉 = 0, the function f ;J captures information that is complementary to thatcaptured by the�-part in the sense that the oscillations are better captured by .There is, however, no heuristic rule for the determination of the parameter½.

In the next section we present an estimate of the error arising due to a finite choiceof J.

4. Error estimate

As noted above, finding proper values ofJ is an important aspect to be dealt with.One has to balance the conflicting effects on the approximation error of the small value


of J and the spread of frequencysupport and consequent dominance of high frequencynoise for large values ofJ. In this section we provide an estimate which helps in somemeasure to choose the proper values forJ. We assume that the function� besidessatisfying the aforementioned properties has support in[−a;a] and possessesM zeromoments, that is, ∫

R

si�.s/ds = 0; i = 1;2; : : : ;M:

For t ∈ [−R; R], let E.t/ be the error arising due to the approximation of.3� f /.t/by [.3� f / ? �J ].t/. Then taking a Taylor series expansion up toN terms (whereN − 1 ≤ M) and using the zero moment property of� we have

E.t/ = [.3� f / ? �J ].t/ − .3� f /.t/

=∫R

[.3� f /.t − s/− .3� f /.t/

]�J.s/ds

[using

∫R

�.s/ds = 1

]=

∫R

[N−1∑l=0

.−s/l

l ! @ l .3� f /.t/ + .−s/N

N! @N.3� f /.cs;t /− .3� f /.t/

]�J.s/ds

=∫R

.−2−Js/N

N! @N.3� f /.cJs;t /�.s/ds =

∫ a

−a

.−2−Js/N

N! @N.3� f /.cJs;t /�.s/ds;

where we have used that supp� = [−a;a] andN − 1 ≤ M . Taking the modulus onboth sides of the above equation, we have

|E.t/| ≤ 2−J N

N!∫ a

−a

∣∣@N.3� f /.cJs;t /

∣∣|sN�.s/| ds

≤ 2−J N

N! ‖.·/N�‖L1.R/ sup[−R−a;R+a]

|@N.3� f /|: (4.1)

Since.3� f /.s/ = ∫R

\.R� f /.t/|t |e2³ i ts dt, we have

@N.3� f /.s/ =∫R

\.R� f /.t/|t |.2³ i t /N e2³ i st dt

and hence

|@N.3� f /.s/| ≤ .2³/N∫R

∣∣\.R� f /.t/∣∣|t |N+1 dt: (4.2)

Now, using (4.1), (4.2) and (2.4) we get∫ ³

0

|E.〈x;u�〉/| d� ≤ 2−J N

N! ‖.·/N�‖L1.R/.2³/N

∫ ³

0

∫R

∣∣\.R� f /.t/∣∣|t |N+1 dt d�

= 2−J N

N! ‖.·/N�‖L1.R/.2³/N

∫R2

| f .¾/|‖¾‖N d¾:


Now, estimating the last integral in the last line of the above inequality and using (2.1),we get∫R2

| f .¾/|‖¾‖Nd¾ ≤∫R2

| f .¾/|.1 + ‖¾‖2/N=2d¾

≤∫R2

| f .¾/|.1 + ‖¾‖2/.N+1/=2+ž

.1 + ‖¾‖2/1=2+ž d¾

≤(∫

R2

| f .¾/|2.1 + ‖¾‖2/N+1+2žd¾

)1=2(∫R2

d¾

.1 + ‖¾‖2/1+2ž

)1=2

=√³

2ž‖ f ‖H N+1+2ž.R2/:

In the above inequalityž is any positive quantity. In particular, when we takež = 1=2,we get

‖ f − f�;J‖L∞ = ess supx∈R2

∣∣∣∣∫ ³

0

E.〈x;u�〉/d�

∣∣∣∣≤ 2−J N

N! ‖.·/N�‖L1.R/.2³/N√³‖ f ‖H N+2.R2/:

Thus we conclude that

‖ f − f�;J‖L∞ = O

(2−J N

N!)

provided f ∈ H N+2.R2/: (4.3)

REMARK 3. Note that the functionf�;J given in (3.2) can be written as

f ≈ f�;J = f ? 8J = <#.< f ? 3�J/; (4.4)

where<# is the adjoint of the Radon transform operator< and8J stands for<#3�J .It may be observed that8J.¾/ = �J.‖¾‖/ and hence8J satisfies

∫R2 8

J.x/dx = 1and zero moments as�. Consequently, estimates of the type

‖ f − f ? en ‖H r .R2/ ≤ C 3‖ f ‖H r +3.R2/ (4.5)

hold as shown in [16, Example 3.4]. In (4.5), en is a radial mollifier defined as

.1= 2/en.1= / for > 0, where the functionen has zero moments as shown in [16,Example 3.4].

The estimate proved in (4.5), like the estimate (4.3) proved in this section, showsfaster convergence providedf has a sufficient degree of smoothness. TheL∞ errorpresent in (4.3) captures the localised error better than any other error estimate. Forexample, any slight shift in the pixel values of the reconstructed image is betteridentified by theL∞ error than by the error involving a Sobolev norm in (4.5).


Generally in applications, we reconstruct pictures that are smooth except for somejumps and such simple pictures relate to density functions inH 1=2 [14], the Sobolevspace of order 1=2. However, in view of the above estimates to achieve faster conver-gence, one requires a higher degree of smoothness forf and sufficiently many zeromoments of�. From (4.3), it is evident that higher order smoothness forf (that is,larger values ofN) can compensate the lower values ofJ, in order to achieve a similarerror estimate. Since the function� has higher order zero moments and is sufficientlysmooth, it decays faster in the frequency domain as well, and as a result� acts as alow pass filter. Consequently, a smallerJ may be helpful in interpolating the datafunction with fewer samples.

An estimate of type (4.3) holds even when we use� = � + ½ in (3.2) with�; satisfying the following zero moment conditions:

∫R

si�.s/ds = Ži;0 and∫R

si .s/ds = 0, i = 0;1; : : : ;M .

5. Examples ofφ

It is well known (see [6]) that coiflets and their corresponding scaling functionssatisfy zero moment properties. It is also known [14] that the Hilbert transforms ofsufficiently regular functions possessing zero moments of sufficient order do not showspread in support. Similarly if wavelet functions of sufficient regularity satisfy thezero moment property, then the supports ofH and are not essentially different.However, the scaling functions have unit mean value and hence do not satisfy thisproperty in general. Nevertheless, for certain classes of scaling functions, the function3� is approximately finitely supported (in the sense that3� takes negligible valuesoutside some finite interval) as documented in [15]. The interval over which3� takessignificant values is called the essential support of3�. In computation the essentialsupport is normally taken based on some error criterion, which is discussed in latersections. The coiflet scaling functions satisfy the stated desirable properties.

Although the coiflet scaling function is a suitable candidate for�, it lacks closedform. If one wants to use� with a closed form, one can construct it in the followingway. Since the procedure given in the present work is not a wavelet-based methodand one can use any�, satisfying the conditions assumed for them, it should beremarked that the use of the part is not a requirement of the procedure. In thefollowing, we present the construction of a� having closed form.

We start with symmetric, compactly supported and sufficiently smooth functionsSi , for i = 1; : : : ; L, for someL. We define the desired function� to be

� =L∑

i =1

ci Si : (5.1)


(a)

8

6

4

2

0

−2

−4420−2−4

(b)20−2

20

15

10

5

0

−5

−10

(c)50−5

2:5

2

1:5

1

0:5

0

−0:5

−1

−1:5

FIGURE1. Thick line forS1, ‘−−’ line for 3S1, continuous line for� and ‘− · −’ line for 3� in both thecases where the� used is based on (a) (5.3), (b) (5.4) and (c) a coiflet (coif3). Vertical lines represent thesupport margins of� andS1.


TABLE 1. Coefficientsci in � and the spread of the function outside the support after the3 operation indifferent cases.

Cubic SplineL = 1 L = 3 L = 4 L = 5

Coeff. 1:0 1:0683 −0:7219 0:4286.ci / −10:9493 16:0089 −17:2165

17:3653 −68:5788 142:896572:8282 −376:4922

300:7969Spread 19:6 3:4363 2:1845 1:6030

PolynomialL = 1 L = 3 L = 4 L = 5

Coeff. 1:23 1:3747 −0:8592 0:4564.ci / −10:4066 14:9595 −15:5829

12:3246 −47:7579 98:500737:9638 −194:5645

116:5074Spread 15:8 2:9277 1:9165 1:3928

The constantsci are determined from the following moment conditions:∫R

x2i�.x/dx = Ži;0; i = 0; : : : ; L − 1: (5.2)

In (5.2), we consider only the even order zero moments as the odd order moments aretrivially zero for symmetric�. Now the function� is compactly supported, smoothand has higher zero moments up to order 2L − 2. As examples, we considerS1 to bethe cubic B-spline (5.3) and a finitely supported polynomial-like function (5.4):

S1.t/ =

t2=2; −3=2 ≤ t ≤ −1=2;

−.2t2 − 6t + 3/=2; −1=2 ≤ t ≤ 1=2;

.t2 − 6t + 9/=2; 1=2 ≤ t ≤ 3=2;

0; elsewhere;

(5.3)

S1.t/ ={.1 − t2/4; −1 ≤ t ≤ 1;

0; elsewhere:(5.4)

It may be observed that in (5.4) as an example we have considered the fourth degreein the definition ofS1 to ensure higher order smoothness forS1. In both cases, weconsiderSi = Si

1 for i = 1;2; : : : ; L. In order to show the usefulness of these


functions for our purpose, we compute the spread of energy of3� outside the supportof � as a fraction of total energy expressed in percentage terms:

‖3�‖L2.R−Supp�/

‖3�‖L2.R/

× 100:

These are tabulated in Table1 along with the correspondingci for L = 1;3;4;5.From Table1, it can be concluded that the spread decreases as the number of zero(higher) moments of� increases. The spread corresponding to the coiflet wavelet(coif3) is 0:8080%. In view of the above, we conclude that any one of these functionscan be used for the purpose of local reconstruction.

6. Error analysis for the localisation of data

As we have stated at the beginning of this paper, our objective in the present workis to reconstruct the interior portion or ROI using localised data (that is, the data fromthat interior region plus some neighbouring data from outside the ROI). The regionfrom which the data is collected and used for interior reconstruction is called theRegion-Of-Exposure or ROE. The present section deals with finding the ROE for agiven ROI.

The estimate proved in Section4 is an estimate for determining the approximationerror when finiteJ is in use and which works only for full data. However, we need anestimate of the error within the ROI as the ROE increases. In this section, we find anupper bound for the error incurred due to the localisation of the data to a small ROE inthe reconstructionof the image in the ROI. Letri andre denote the radii of discs centredat the origin of the ROI and ROE respectively. Let[−a;a] be the essential supportof the functions3� and3 (note that3� is symmetric whenever the function� is).Then the extent of data needed to recover the local region isre = ri + a=2J, whichcorresponds to the maximum possible overlap of supports of.R� f / and.3�/J with‖x‖ ≤ ri . Analogous consideration holds for the part involving . In the followingwe prove an estimate that relates the error arising due to the use of localised data andthe extent of localised data (re). From (3.3), we have

f�;J.x/ =∫ ³

0

∫R

R� f .t/3�J.〈x;u� 〉 − t/dt d�

=∫ ³

0

(∫|t |≤re

+∫

|t |>re

)R� f .t/3�J.〈x;u� 〉 − t/dt d�:

Let us define the error functionE�;J.x/ to be

E�;J.x/ = f�;J.x/−∫ ³

0

∫|t |≤re

R� f .t/3�J.〈x;u� 〉 − t/dt d�: (6.1)


Using the fact thatf has compactsupport in the disc of radiusR, |R� f .t/| ≤ 2R‖ f ‖∞.Whenx belongs to the circle centred around the origin of radiusr i , |〈x;u�〉| ≤ ri .From the definition of3,3�J.t/ = 22J3�.2J t/. Using all these relations, we get

|E�;J.x/| =∣∣∣∣∫ ³

0

∫R>|t |>r e

R� f .t/3�J .〈x;u�〉 − t/dt d�

∣∣∣∣≤ 2³R‖ f ‖∞

∫ ³

0

∫R>|t |>r e

|3�J.〈x;u� 〉 − t/| dt d�

= 2³R‖ f ‖∞

∫ ³

0

∫R>|t−〈x;u� 〉|>re

|3�J.t/| dt d�

≤ 2³R2J‖ f ‖∞

∫ ³

0

∫2J .R+r i />|t |>2J.re−r i /

|3�.t/| dt d�

≤ 2³2 R2J‖ f ‖∞

∫|t |>2J.re−r i /

|3�.t/| dt: (6.2)

In (6.2), the fourth step follows from the third step due to the following reasons. Therelations|〈x;u�〉| ≤ ri andR> |t −〈x;u� 〉| > re imply R+ r i > |t | > re − ri . Since{t : R > |t − 〈x;u�〉| > re} ⊂ {t : R + ri > |t | > re − ri }, the inequality symbolfollows in the fourth step. Using the relation3�J = 22J3�.2J t/ and replacing 2Jtwith t , we get the fourth step. Finally, with� in the place of�, we have

‖E�;J‖L∞.ROI/

R‖ f ‖L∞.R2/

≤ 2³22J

∫|t |>2J.re−r i /

|3�.t/| dt: (6.3)

Observe that the right-hand side of the above inequality is independent off . From theabove inequality, it may be concluded that at a givenJ, asre increases, the right-handside of (6.3) decreases. The decay of the right-hand side of (6.3) against the increasein .re − ri / is shown in Figure2 using the functions� constructed based on (5.3), (5.4)at L = 3 (5.1) and the coiflet (coif3). From the graphs, which are independent ofr i

and the test imagef , it may be concluded that whenre = ri +a=2J for a > 4:5 (in thefirst case),a > 3 (in the second case) anda > 10 (in the third case), the error arisingout of localisation of data to the interior region becomes exceedingly small. There isyet another type of error due to the discretisation of data. It is, however, quite standardto take care of it by sampling the data using Nyquist sampling rates (as dictated by theessential band limit of�).

Although from the above error analysis we get information about the radius of theregion of exposure, in the simulation part, we start with a� with support[−a;a](say). Then we assume that3� has essential support[−¹; ¹] with increasing valuesof ¹ starting witha. We see how the error and quality of the reconstructed local imagechange in the course of simulation.


(a) (b) (c)

000

51010

10

152020

20

25

3030

30

35

4040

40

5050

6060

11 222 33 44 6 8

FIGURE 2. Plot between.re − r i / and the right-hand side of (6.3). The function� used is based on(a) (5.3), (b) (5.4), and (c) a coiflet (coif3). In all three cases, we use a continuous line forJ = 3, a ‘−·−’line for J = 2, and a ‘− −’ line for J = 1.

7. Some comparisons

Although the algorithm presented in this section is not wavelet based, nevertheless,a comparison of it with the algorithm promoted by Faridaniet al. in [8, 9, 10] seemsworthwhile as both the methods in some way use approximation identities for certainfunctions. The3F operator

3F f .x/ = .−1/1=2 f .x/

= − 1

4³1<#< f .x/ = − 1

4³

∫ 2³

0

@2.R� f /.〈x;u� 〉/d� (7.1)

studied by Faridaniet al. is local and is aimed at capturing a certain high frequencypart of f . In (7.1), 1 represents the Laplace operator. According to the authors of[8, 9, 10], in practical applications of3-tomography, they do not compute3F f , butrather attempt to reconstruct3F .�ž ? f /.x/ [13, 17] for some approximation identity�ž.x/ = ž−2�.x=ž/, ž > 0, where� is an integrable function with total integral one.Now for a good choice ofž, 3F f .x/ is computed via

3F f .x/ ≈ 3F.�ž ? f /.x/ = [ž−1.3F�/ž ? f ].x/= .ež ? f /.x/ = <#.vž ?< f /: (7.2)

In the above,ež = ž−1.3F�/ž andvž = 3<ež . Finally, ež ? f is computed via theback projection method. In the end, the image that one gets is not an approximationof the original image, but of the high frequency part off . To include the missing low


frequency part of3F f , another operatorL f given by

L f := 3F f + ¼3−1F f (7.3)

has been studied and analysed in [9] and [10]. The operator3−1F included in L

represents the inverse of the operator3F and (see [13]) is given by

3−1F f .x/ = 1

4³<#< f .x/ = 1

4³

∫ 2³

0

.R� f /.〈x;u� 〉/d�: (7.4)

Here again the functionL f is a local function which does not representf but arelated function [10]. The authors of [9] and [10] have studied and demonstrated theeffects of3F and3−1

F and have considered the coefficient¼ by trial and error in theirexperiments. Although we also introduce a parameter½ with a view to enhancingthe reconstructed image, we observe that the role of½ is marginal when we considerlarger values ofJ. We discuss this aspect in the section dealing with simulation. Butmost importantly, the inclusion of½ (or the -part) is not mandatory in our procedurefor reconstruction. Hence no question of comparison between the roles of½ in (3.4)and¼ in (7.3) arises.

The method established by A. K. Louis [11] uses the concept of approximateidentity for the purpose of regularisation. In [11], the method involves solving theequationR#v = e for a mollifier (approximate identity)e with

∫e.x/dx = 1 by

v = .2³/−13Re. The method is then given as

f .x/ = R f ? D ;xv;

whereD ;xv.�; t/ = −2v.�; .t − 〈x;u�〉/= /.Although the method proposed in the present work uses the properties of approx-

imation identities of certain classes of functions, it differs in its objective from theother algorithms using the same concept. The basic intention in the present work isthe reconstruction of the original local image as accurately as possible using local databy using a CBP like procedure.

The approximate computation of3F f and the estimation of associated approxima-tion errors have been the subject of study by Riederet al. in [17]. The basic objectiveof the authors in [17] is to investigate the properties of a compactly supportedv, wherev = 3<e in (7.2) and3F f ≈ f ? ež. An error estimate is derived by taking a zeromoment condition forv for even integers (that is,

∫s2kv.s/ds = 0, for k = 0;1).

In (4.4), 3� plays the same role asv in (7.2). Although, for symmetric�, 3�satisfies the same zero mean and zero moment conditions asv, it lacks compactsupport. Hence the estimate given in [17] is not applicable in the present context.

Basically, the method given here involvesone leveland the methods involvingwavelet decompositions usea hierarchy of levels[5, 7, 15]. In addition the wavelet-based decompositions involve translations which also add to the computations.


(a) (b) (c)

FIGURE 3. (a) f�;0, (b) 5f ;0 and (c) f�;0 + f ;0.

(a) (b) (c)


(a) (b) (c)


(a) (b)

FIGURE6. (a) f�;5 + f ;5 and (b) original test image.


8. Results of simulation

As stated earlier, a wide variety of� and can be used in simulation. Since theinclusion of is not at all compulsory in (3.4), one can carry out the simulations usingthe� functions alone constructed in Section5. However, we present in the followingthe results obtained using the coiflet pair� and and study the usefulness of the -part when the considered is orthogonal to�.

We have carried out computations using the coiflet (‘coif3’) wavelet and scalingfunctions for differentJ. Reconstructions of the Shepp-Logan head phantom havebeen executed on 256×256 pixel grids using 256 projections collectedat 256 uniformlyspaced angles over[0; ³/ and the reconstructed images are shown in Figures3–6. Itmay be observed that these figures are displayed by discarding the portion lying outsidethe outer ellipse to make the changes occurring in the main part more visible. In allof Figures3–5, the wavelet part is shown with½ = 5 to emphasise the nature of thecontribution of f ;J to f . We have computed theL∞, L2 errors in % terms using theformula

L p error ( in %/ = ‖ f − fr‖p

‖ f ‖p

× 100; p = 2;∞; (8.1)

for the cases� = � as well as� = � + (that is,½ = 1). Here, fr stands for thereconstructed image. The errors computed are shown in Table2. It may be observedfrom Table2 that the part does not change the errors significantly forJ ≥ 3 andthe L∞ error comes below 1% forJ ≥ 6.

TABLE 2. L p, p = 2;∞, errors in % with�, � + in place of� at different values ofJ.

J L∞.%/ : � L∞.%/ : � + L2.%/ : � L2.%/ : � +

0 221.36 221.23 79.67 75.131 232.34 230.53 44.84 44.142 131.09 130.71 12.71 12.643 47.187 47.179 1.618 1.6174 12.945 12.945 0.124 0.1245 3.3125 3.3125 0.008 0.0086 0.8330 0.8330 5:197× 10−4 5:197× 10−4

7 0.2085 0.2085 3:2591× 10−5 3:2591× 10−5

8 0.0522 0.0522 2:0386× 10−6 2:0386× 10−6

9 0.0130 0.0130 1:2744× 10−7 1:2744× 10−7

10 0.0033 0.0033 7:9655× 10−9 7:9655× 10−9

When the ‘coif3’ wavelet is in use for local reconstruction, the excess (outside theROI) data to be used amounts to just 6 pixels only forJ = 0 (that is, the radius of the


TABLE 3. L p, p = 2;∞, errors in % with constant and zero extensions of data outside the ROE atdifferent values ofJ.

J Nonlocal L∞.%/ L∞.%/ L2.%/ L2.%/data(pixels) Const. Ext. Zero Ext. Const. Ext. Zero Ext.

0 6+0 86.556 85.831 46.540 105.8110 6+6 85.309 68.755 43.572 57.2680 6+10 84.295 61.537 41.153 41.9021 3+0 64.612 83.466 31.547 127.1031 3+6 64.115 65.883 30.458 58.2851 3+10 63.169 60.349 28.067 36.6942 2+0 45.349 78.145 25.419 137.1452 2+6 45.279 56.222 25.601 59.67012 2+10 44.304 48.417 23.265 35.4083 1+0 28.813 72.293 22.511 155.9773 1+10 28.379 35.945 21.785 38.1364 1+0 21.564 71.229 22.183 155.2294 1+6 21.769 42.266 23.509 65.48874 1+10 20.751 31.169 21.455 37.7855 1+0 21.092 71.438 22.154 155.0895 1+6 21.211 42.112 23.480 65.4415 1+10 19.911 30.954 21.427 37.7526 1+4 21.404 49.668 23.909 86.5266 1+6 21.139 42.189 23.478 65.4356 1+8 20.593 36.086 22.642 49.6646 1+10 19.839 31.013 21.412 37.7406 1+15 18.192 21.282 18.495 18.8546 1+20 16.552 14.739 15.311 9.0296 1+25 16.602 9.779 14.831 3.930

ROE is the radius of the ROI+9 for J = 0). Lower numbers for the ROE can possiblybe achieved using several other suitable pairs given in [4]. In all the computations,the ROI is the central 1=4 portion of the full image (that is, radius= 32 pixels). Wehave made computations for local reconstructions at different values ofJ. To avoidthe artifacts caused due to the truncation of data to the ROE, we have considered aconstant extension of the data outside the ROE as expressed through the equation

.R� f /.t/ = .R� f /.re/ for all |t | ≥ re and �: (8.2)

Otherwise, the sharp cut off (discontinuities) introduced into the data function mayhave an adverse impact on the quality of the outputs due to Gibb’s phenomenon.However, in computations, we have estimated the errors on the ROI both with theconstant as well as with zero extension of data outside the ROE and the recordings


(a) (b)

FIGURE7. Reconstruction on the ROI of radius 32 pixels atJ = 0, that is,f�;0 with (a) constant extension,(b) zero extension of data outside the ROE.

(a) (b)

FIGURE8. Reconstruction on the ROI of radius 32 pixels atJ = 3, that is,f�;3 with (a) constant extension,(b) zero extension of data outside the ROE.

are shown in Table3. In the second column of Table3, we have shown the nonlocaldata in ‘p + q’ form, where ‘p’ denotes the overlap of supports of.R� f / and theramp function outside the ROI (that is,a=2J in terms of pixels—as stated in the lastparagraph of Section5) and ‘q’ stands for the excess data that we supply to reducethe errors. From Table3 it may be observed that theL∞ error (in % terms) shows avery slow decline in value when we increase the ROE afterJ = 4.

As was pointed out in [12], the interior reconstruction procedures give a constantshift in the absolute value of the density function to be found. We have observedthat a removal of a constant bias with value−0:14 from the reconstructed image atJ = 4 and ROE=11 pixels results in a fall inL∞ error to 3% from nearly 20%. Thereconstructed ROI images shown in Figures7–9 suggest that with an excess marginof 11 pixels one can have good local images. It may be observed that the -part in


(a) (b) (c)

FIGURE9. Reconstruction on the ROI of radius 32 pixels atJ = 4, (a) before removing constant bias and(b) after removing constant bias (c) original local image obtained via the CBP procedure.

(a) (b)

FIGURE 10. 5f ;J at (a)J = −1 and (b)J = −2.

the coiflet case gives negligible contribution tof . We have observed the errors fordifferent J’s and½’s (that is, � = � + ½ ) and our conclusion is that for positiveJ, f ;J ’s role is negligible. However, for negativeJ, f ;J captures edge informationas shown in Figure10. Hence the inclusion of the -part with negativej into therepresentation, that is, with� = �J +½ − j for some appropriatej and½, may enhancethe edges present in the images. Since the choice ofj and½ is case dependent forimage enhancement problems, we will not go into the details of computational workin this regard. It is theoretically justified that the� functions constructed in Section5work well for local reconstructions. We have observed similar results, as presentedin these sections, using the� constructed in Section5. To avoid presenting too manyresults, we omit them here.

9. Conclusion

The algorithm proposed in the present work uses space and frequency localisationproperties of certain functions having zero moments. These functions form an ap-


proximation identity and remain essentially compactly supported after ramp filtering.The algorithm retains the structure of the filtered back projection (FBP) procedure andadmits faster and standard ways of implementation using a relatively small amount ofdata for recovering local regions.

Acknowledgements

The authors thank the referee for his/her suggestions. The first author is grateful tothe Council of Scientific and Industrial Research (CSIR), India for its financial support(Grant No. FNO:9/92 (164)/98-EMR-1).

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